Homework 4



Homework 4: Due Friday 31 January


  1. You have 30 fish in 30 tanks. You want to test and see if the fish, if put together, will fight with each other or get along harmoniously. To do so, you want to put a pair of fish in a special tank and monitor them for a day. Your colleague wants to pair up the 30 fish in each possible way. How many days will it take to complete this experiment?
  2. When a lobsterman catches lobsters, there is a 60% chance that the lobsters will be big, and a 40% chance that the lobsters will be small. If a lobsterman catches 12 lobsters one day, what is the probability that he catches 8 big lobsters?
  3. You have 20 CDs. How many different ways can you arrange them on a shelf?
  4. After an in vitro fertilization procedure (IVF) four fertilized eggs are placed in the mother's uterus. Assume that each egg has one chance in ten of implanting successfully, and that this is independent of whether any of the others implants. What is the probability that none of the eggs will implant? That a single egg will implant? That there will be twins? That there will be triplets? That there will be quadruplets? That there will be qunitrupulets? (From http://www.math.lsa.umich.edu/~hochster/425/ec2.html.)
  5. Consider the complex numbers z_1 = 4 - 4i and z_2 = 2i.
    1. Convert both number into exponential form
    2. Find z_1z_2.
    3. Find z_1z_1^*. (I.e., z_1 times its complex conjugate.)
    4. Find z_2z_2^*. (I.e., z_2 times its complex conjugate.)
Bonus/Challenge Problems/Other things to try
  1. A cube that is painted blue is cut into 64 equal cubes. What is the probability P_n that a little cube picked at random has n painted faces, where n = 0, 1, 2, 3? (From Garrod, Statistical Mechanics.)
  2. Ten people throw dice, once per minute, at ten tables. When any person throws 12 he or she leaves. What is the probability that anyone will be left after one hour? (From Garrod, Statistical Mechanics.)
  3. Note: this is much more difficult than the other problems. For parts 2 and 3, a little bit of calculus is probably necessary. The new breakfast cereal, Millenios, consists of pieces in the two shapes 0 and 2. Thus, a spoonful of these pieces might contain a 2 and the three 0's needed to spell 2000. Suppose that a spoonful of n Millenios is obtained from a machine that produces, independently and randomly, a 2 with probability p and a 0 with probability 1-p. (This situation approximates the selection of a spoonful of n pieces from a large box full of Millenios in which the ratio of 2's to 0's is p:1-p.) (Problem 679 from the College Journal of Mathematics.)
    1. For n=4, what value of p maximizes the probability that the spoonful will contain the 2 and the three 0's needed to spell 2000?
    2. For arbitrary n >= 4, let p_n be the value of p that maximizes the probability that a spoonful of n pieces will contain at least one 2 and at least three 0's. Determine p_n as a function of n.
    3. Find the limit of p_n from part (2) as n goes to infinity. (It is obvious that for an p satisfying 0 < p < 1, as n goes to infinity the probability that a spoonful of n pieces will contain at least one 2 and at least three 0's will approach 1.)
  4. In a certain country the percentages of boy babies and girl babies born are equal under normal circumstances, and their life expectancies are the same. But a tradition develops that families keep having children until a girl is born, and then stop. (Of course, they may have to stop for other reasons.) What is the expected ratio of males to females in the population? (From http://www.math.lsa.umich.edu/~hochster/425/ec2.html.)

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